## Term rewriting for normalization by evaluation

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Venue: | 19–42, International Workshop on Implicit Computational Complexity (ICC’99 |

Citations: | 11 - 2 self |

### BibTeX

@INPROCEEDINGS{Berger_termrewriting,

author = {Ulrich Berger and Matthias Eberl and Helmut Schwichtenberg},

title = {Term rewriting for normalization by evaluation},

booktitle = {19–42, International Workshop on Implicit Computational Complexity (ICC’99},

year = {},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

We extend normalization by evaluation ( rst presented in [5]) from the pure typed-calculus to general higher type term rewriting systems. We distinguish between computational rules and proper rewrite rules, and de ne a domain theoretic model intended to explain why normalization by evaluation for the former is much more e cient. Normalization by evaluation is proved to be correct w.r.t. this model. 1

### Citations

424 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ... -calculi In this section, we shall discuss the domain theoretic semantics of simply typed lambda calculi in general. Although the constructions below are standard (see e.g. the books of Lambek/Scott =-=[10]-=- orCrole [7]), we discuss them in some detail in order to make the paper accessible also for readers not familiar with this subject. Most constructions make sense in an arbitrary cartesian closed cate... |

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...called with a term M such that M \Gamma! Q will terminate with Q as output. In order to obtain operational correctness from denotational correctness one needs a suitable adequacy result 'a la Plotkin =-=[13]-=- relating the denotational and the operational semantics. Plotkin's result cannot be applied here because it refers to a call-by-name operational semantics, whereas we are interested in a call-by-valu... |

346 | Recursive functions of symbolic expressions and their computation by machine (Part I
- McCarthy
- 1960
(Show Context)
Citation Context ...sAll these rules are possible computation rules, whereas the next two arerules are not (since if and add are no constructors). if (if xyz) 7;! uv:if x(if yuv)(if zuv)� (a rewrite rule due to McCarthy =-=[11]-=-), or mult(add xy) 7;! z:add(mult xz)(mult yz): (b) We can also deal with in nitely branching trees like theBrouwer ordinals of type O. There are constructors 0 O and Sup ( !O)! ,andrecursion constant... |

303 | N.: Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation.; Indag
- Bruijn
- 1972
(Show Context)
Citation Context ...:= xk(# ; a(" (x1 k )) (k +1)), where x1k is the constant family xk. 2sThe de nition of " ! has to be modi ed accordingly. This idea corresponds to a representation of terms in the style of de Bruijn =-=[9]-=-. An advantage of this approach isthat the NbE program is purely functional and hence can be veri ed relatively easily. If side e ects were involved the veri cation would be much more complicated. The... |

207 | Type-directed partial evaluation
- Danvy
(Show Context)
Citation Context ...ch has the property that the interpretation of the simply typed lambda calculus in it yields the reduction-free normalization algorithm from [5], as well as its correctness. Moreover, Danvy (cf. e.g. =-=[8]-=-) has successfully used this algorithm in the context of compiler construction. The aim of the present paperis to develop the theory of normalization by evaluation from scratch, up to and including (s... |

80 |
Categories for Types
- Crole
- 1993
(Show Context)
Citation Context ...this section, we shall discuss the domain theoretic semantics of simply typed lambda calculi in general. Although the constructions below are standard (see e.g. the books of Lambek/Scott [10] orCrole =-=[7]-=-), we discuss them in some detail in order to make the paper accessible also for readers not familiar with this subject. Most constructions make sense in an arbitrary cartesian closed category (ccc). ... |

50 |
An inverse of the evaluation functional for typed -calculus
- Berger, Schwichtenberg
- 1991
(Show Context)
Citation Context ...mplementing normalization of -terms in the usual recursive fashion is quite ine cient. However, it is possible to compute the long normal form of a -term by evaluating it in an appropriate model (cf. =-=[5]-=-). When using for that purpose the built-in evaluation mechanism of e.g. Scheme (a pure Lisp dialect) one obtains an amazingly fast algorithm called \normalization by evaluation" or NbE for short� a c... |

44 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...will certainly bene t from mathematical studies like the present one, but the details still have tobedone. Arelated approach (using a glueing construction) is elaborated by T.Coquand and P. Dybjer in =-=[6]-=-. Another related paper is T. Altenkirch, 1sM. Hofmann and T. Streicher [1]� there a cartesian closed category is dened which has the property that the interpretation of the simply typed lambda calcul... |

25 | Proof theory at work: Program development in the Minlog system
- Benl, Berger, et al.
- 1998
(Show Context)
Citation Context ...ntial idea is to find an inverse to evaluation, converting a semantic object into a syntactic term. This normalization procedure is used and tested in the proof system Minlog developed in Munich (cf. =-=[2]-=-). -- Notice, however, that once NbE is expressed in a functional programming language, the evaluation order of this language (call-by-value for Scheme) determines the reduction order of NbE (applicat... |

25 | A semantic account of type-directed partial evaluation
- Filinski
- 1999
(Show Context)
Citation Context ... as well as its correctness. Moreover, O. Danvy (cf. e.g. [8]) has successfully used this algorithm (or more precisely its call-by-value counterpart) in the context of partial evaluation. A. Filinski =-=[10]-=- also treats NbE for an extension of the -calculus by constants, where non-termination is allowed. However, he does not consider constants whose meaning is only given operationally, i.e. by arbitrary ... |

22 | Categorical reconstruction of a reduction free normalization proof
- Altenkirch, Hofmann, et al.
- 1995
(Show Context)
Citation Context ...he details still have tobedone. Arelated approach (using a glueing construction) is elaborated by T.Coquand and P. Dybjer in [6]. Another related paper is T. Altenkirch, 1sM. Hofmann and T. Streicher =-=[1]-=-� there a cartesian closed category is dened which has the property that the interpretation of the simply typed lambda calculus in it yields the reduction-free normalization algorithm from [5], as wel... |

13 |
Helmut Schwichtenberg. Normalization by evaluation
- Berger, Eberl
- 1998
(Show Context)
Citation Context ...alization procedure is used and tested in the proof system Minlog developed in Munich (cf. [2]). Obviously,for applications pure typed -terms are not su cient� one clearly needs constants as well. In =-=[4]-=- NbE has been extended to term systems with higher order term rewrite rules. The present paper adds a distinction between what we call computational rules and (proper) rewrite rules� NbE seems to be m... |

9 | Continuous functionals of dependent and transfinite types. In Models and computability
- Berger
- 1997
(Show Context)
Citation Context ...ition of a family of domains (I()) i.e. a least fixed point of certain continuous functions. The theory of (continuous) families of domains and recursive definitions thereof is developed in detail in =-=[3]-=-. 3 From a mathematical point of view it is also possible to take the coalesced sum, but the identification of an undefined object with the total undefined term family is computationaly doubtful. 4 It... |

1 |
Proof theory at work: Program development inthe Minlog system
- Benl, Berger, et al.
- 1998
(Show Context)
Citation Context ...sential idea is to nd an inverse to evaluation, converting a semantic object into a syntactic term. This normalization procedure is used and tested in the proof system Minlog developed in Munich (cf. =-=[2]-=-). Obviously,for applications pure typed -terms are not su cient� one clearly needs constants as well. In [4] NbE has been extended to term systems with higher order term rewrite rules. The present pa... |

1 |
Continuous functionals of dependent and trans nite types. Habilitationsschrift, Mathematisches Institut der Universitat Munchen
- Berger
- 1997
(Show Context)
Citation Context ... nition of a family of domains (I( )) i.e. a least xed point of certain continuous functions. The theory of (continuous) families of domains and recursive de nitions thereof is developed in detail in =-=[3]-=-. 3 From a mathematical point ofviewitisalsopossible to take thecoalesced sum, but the identi cation of an unde ned object with the total unde ned term family is computationaly doubtful. 4 It is easy ... |